1. Technical Field
The subject matter includes a procedure that can be incorporated into Finite Element Analysis (FEA) or other similar analysis techniques, to obtain the steady state temperature distribution in a coupled transient heat transfer analysis rapidly as well as accurately.
2. Background Art
In 1971 Tay and Stevenson made use of the finite element method for evaluation of the temperature distribution along the chip tool interface. They carried out a heat transfer analysis with the actual geometry and size of the chip, workpiece and tool. The heat sources in both the primary and the secondary shear zone were calculated from the strain rate and strain distribution obtained experimentally, using an assumed flow stress model. They had used temperature dependent thermal properties. The strain rate and velocity distributions were adopted from Stevenson and Oxley's work, where an explosive quick-stop device was made use of to evaluate the deformation of a lithographically applied grid on the workpiece. The flow stress at a point was determined as a function of strain, strain rate and temperature. They observed that it was more difficult to get the velocity and strain rate distribution in the secondary shear zone region than in the primary shear zone region. This was attributed to the fact that in the secondary deformation region, there is very intense distortion in a very narrow region and also the printed grid itself is severely distorted in the primary shear zone before entering the secondary shear zone region. They computed the gradients of the velocity distribution to obtain the strain rates. To get the flow stress in the secondary shear zone region, they made use of the measured frictional force. The heat input everywhere was the product of the flow stress and the shear strain rate. In this way they obtained the temperature distribution in the regions of interest.
In their previous work, getting the velocity fields from the printed grid method and then applying it to the FEA model was a very laborious job and was prone to inaccuracies. Also, it required substantial computation time.
Yen et al. (2004) carried out a multi step finite element simulation using the DEFORM 2D FEA package. The first step of the analysis was the thermo-viscoplastic Lagrangian cutting simulation. The steady chip geometry was obtained in this first stage of analysis. In order to achieve the thermal steady state a second stage analysis which was a pure heat transfer analysis was carried out for the tool. The contact heat flux along the chip tool interface acted as the heat input. While the contact heat flux could be expected to decrease as the tool temperature increases, this was not taken into consideration. Thus the steady state tool temperatures were obtained at the end of second stage of the analysis. These results were used in the calculation of tool wear based on Usui's wear model, in the third stage of the analysis. The worn tool geometry obtained at the end of the third stage of the analysis was again used to carry out a fourth stage analysis, which was again a thermo-viscoplastic analysis. This loop of analyses was carried out in order to get the continuous tool wear.
The PhD work by Yaacov Krispin (1987) deals with finding steady state solutions for chemically reacting non-equilibrium flow fields. The steady state solution is difficult to obtain since the governing equations are very stiff. This leads to a very small timestep size as has been identified to be the problem in our current work also. However, in this PhD dissertation, the acceleration to the steady state solution (by almost 10×) has been achieved by replacing the source terms of the original system of equations (which are responsible for the stiffness) by approximate source terms. This is not only different compared to our approach of reducing the thermal inertia but also has the limitation that it can be applied only to thermochemical coupling. Our novel method can be applied to thermal phenomenon coupled with any other phenomenon.
In the 1996 work of Cheng et. al., a new and fast thermal reliability diagnosis tool (iTHREAD) for CMOS VLSI chips has been presented. It is claimed that this tool can not only obtain steady state temperature distribution but also the hot spots, as well as the resulting power consumptions. This work uses a fast timing simulator with accurate temperature-dependent device models and a novel 3-D analytical thermal simulator. The method described in this paper is analytical and doesn't involve finite element analysis. Another difference compared to our scaled specific heat approach is that the method in this paper can be used only for thermal analysis coupled with electrical analysis. Whereas, our scaled specific heat approach can used for thermal analysis coupled with any other phenomenon chemical or electrical with only the minor modification of choosing the scale factor appropriately based on the maximum time step size for accurate calculations of the other phenonmena.
Another work by Tony A Asghari of Motorola Inc. in 2002 claims to accelerate attainment of steady state thermal solutions. The approach used in this work is that, instead of running a full Computational-Fluid-Dynamics (CFD) analysis which involves solving for mass, momentum, and energy equations using finite volume (which is computationally very expensive and needs huge amounts of disk storage space), only the energy equation is solved, incorporating into this the heat-transfer coefficients (h) determined from full CFD steady-state runs at various power-dissipation levels (i.e. full coupling between the flow field and thermal field is replaced by weak coupling). It is claimed that this simplified heat-transfer-coefficient CFD model, or h-model significantly reduces computation time for steady-state problems from 4.5 hours to a few minutes and can be customized as per the customer required power scenarios. This work acknowledges that the temperature profiles obtained with their new method is within 5% of the temperature distribution obtained from the full fledged CFD simulation. While, this approach reduces the simulation time, it is different from what has been demonstrated in our scaled specific heat approach, with the approach described by us here being easier to implement, as well as resulting in accurate solutions at steady state; since the coupling of the thermal phenomenon with flow, electrical, mechanical, etc., phenomena is not affected, their interdependence is also captured accurately.
In his dissertation work, Mikhail Noskov (2003) has demonstrated the use of a stable fully implicit compact scheme solver for monitoring steady state heat generation in multicomponent reacting mixtures. The governing equations in this work are based on the vorticity-velocity formulation. The solution method used is the damped modified Newton's method. Based on the results, it has been shown that the new implicit method leads to significant reduction in computational time compared with the first order accurate implicit solver. A sample calculation showing the heat generation in a laminar methane-air diffusion flame with detailed chemical kinetic and transport model has been presented. While this work claims that the computation time is significantly reduced for a coupled thermal phenomenon, this has been achieved by a new implicit solution approach and not by reducing the thermal inertia (specific heat capacity) as shown in our current work. Also, this dissertation work focuses only on the thermochemical coupling, whereas our current work is more generally applicable.
In the 2005 paper of Qi et. al., a novel approach to study the dynamic thermal management in an integrated circuit has been demonstrated. This study has been carried out at the electronic chip level and is based on the observation that the power consumption of architecture level modules in microprocessors shows strong periodicity. The problem has been broken down into two steps; 1) the steady state response obtained by performing a fast spectrum analysis in frequency domain and 2) The transient temperature changes due to initial conditions and constant power inputs obtained by using moment matching approach which is numerically stable. The final output is the sum of the outputs of these two steps. It has been claimed that this fast thermal analysis algorithm leads to 10× to 100× speedup over traditional integration-based transient analysis. While this work acknowledges that there is a small loss in the accuracy of the solution, it is also different compared to our approach of reducing the thermal inertia of the body. This work performs spectrum analysis in the frequency domain whereas our approach is used in the time domain itself. Also, there is no loss of accuracy using our approach of scaling the specific heat.
In the work of Celo et. al. in 2005, a model reduction technique has been used to perform the thermal analysis of electronic components and devices which have complex geometries. It has been claimed that this model with a reduced order has the capability to predict a detailed 3 dimensional steady state temperature distribution with an accuracy of 0.1% compared to a detailed numerical model. It is also mentioned that the small size and the simplicity of the reduced model helps in performing the simulation very quickly as well as under very wide ranges of input parameters like different boundary conditions and power distributions. Although, very high accuracy has been claimed in this work, reducing the size of the model may have ‘size’ effects on some other thermal problems. So this method can only be used to study the specified electronic components. On the other hand, the new scaled specific heat capacity method demonstrated by us doesn't have any restriction on the type of problem to which it can be applied and is more simple to implement compared to the reduced model size method mentioned in this paper.
In the 2006 paper of Yang et. al., the improvement in the thermal analysis performance and accuracy in integrated circuits has been demonstrated. It is claimed that the method in this paper accelerates the steady state solution by 21.6 to 690 times compared with the conventional analysis techniques. The new method here is composed of spatially-adaptive multigrid iterative solver, a new temporally and spatially adaptive asynchronous time marching solver, and a new spatially-adaptive frequency-domain moment matching solver. Together, these analysis techniques allow the solution system to efficiently perform the static, short time scale, and long time scale variants of the IC thermal analysis problem. Again, in this work the acceleration to the steady state solution has been achieved, but, by modifying the solver algorithm itself. This is different compared to our work, where only the thermal inertia of the body is reduced so that the steady state can be achieved faster. In our work, there is no change to the solution algorithm itself. Our approach is much easy to implement and is independent of any software or solver algorithm. The work in this paper can be benefitted immensely by our scaled specific heat capacity analysis technique.
In their patent (#WO 1995-019007 A1), Bouchez et. al. have demonstrated a method of predicting steady state condition based on the transient monotonic or the cyclic data from previous time. They have identified a first and second property state value corresponding to the transient data at specific times within the testing time period. Based on the first and second rate-of-change value a time constant value is computed next. In order to obtain the projected steady state condition value, the second rate-of-change value, the second state value and the time constant value are used with a projected steady state conditions function. This process is then repeated until the difference between two steady state condition values is then less than a threshold value. This approach is similar to fitting a curve to the available transient data and then extrapolating it ahead in the time until the slope of the curve approaches zero (i.e. steady state is attained). The drawback of this approach is that the steady state obtained is based on data extrapolation and it cannot be said with certainty if it will be the actual steady state of the system. Also, the steady state mentioned here doesn't correspond to thermal steady state but is a general term for a system under environmental and/or operating conditions.
In their patent numbered WO 2007-070879 A1, Chandra et. al. have demonstrated the use of heuristics to adaptively gird space in 3 dimensions. This approach is used to model the temperature distributions in the integrated circuit chip. This locally variable gird has been used to obtain transient and or steady state temperatures. If localized high temperature regions are observed, the thermally aware design suit will automatically make a change to the chip layout so as to make the temperature distribution uniform. The main aim of this work is significantly different compared to the present work of scaled specific heat capacity. Also, the steady state temperatures are obtained in a different manner using heuristics.
In the patent # U.S. Pat. No. 4,259,859 A by Iida et. al., a method by which the thermal properties like specific heat, thermal conductivity, and thermal diffusivity of a system involving basically and perfectly arbitrary boundary and heating conditions with ease of operation and high accuracy has been described. This work relates just to obtaining the thermal properties of the material and doesn't involve any thermal steady state analysis or coupled thermal steady state analysis. Thus, this work is different than what has been proposed in the present scaled specific heat capacity approach of obtaining the thermal steady state in a coupled thermal phenomenon.